Elastic Beam Equation Research Articles

Existence and multiplicity of positive solutions for an elastic beam equation

This paper investigates the boundary value problem for elastic beam equation of the form $$u''''(t) = q(t)f(t,u(t)u'(t),u''(t),u'''(t)),0 < t < 1,$$ , with the boundary conditions $$u(0) = u'(1) = u''(0) = u'''(1) = 0.$$ . The boundary conditions describe the deformation of an elastic beam simply supported at left and clamped at right by sliding clamps. By using Leray-Schauder nonlinear alternate, Leray-Schauder fixed point theorem and a fixed point theorem due to Avery and Peterson, we establish some results on the existence and multiplicity of positive solutions to the boundary value problem. Our results extend and improve some recent work in the literature.

Non-trivial solutions for nonlinear fourth-order elastic beam equations

The existence of at least one non-trivial solution to a boundary value problem for fourth-order elastic beam equations, under a non-standard growth condition of the nonlinear term, is established. Our approach is based on a local minimum theorem for differentiable functionals.

Infinitely many solutions for a fourth-order elastic beam equation

Existence results of infinitely many solutions for a fourth-order nonlinear boundary value problem are established. No symmetric condition on the nonlinear term is assumed. The main tool is an infinitely many critical points theorem.

Eigenvalue Problem and Unbounded Connected Branch of Positive Solutions to a Class of Singular Elastic Beam Equations

This paper investigates the eigenvalue problem for a class of singular elastic beam equations where one end is simply supported and the other end is clamped by sliding clamps. Firstly, we establish a necessary and sufficient condition for the existence of positive solutions, then we prove that the closure of positive solution set possesses an unbounded connected branch which bifurcates from Our nonlinearity may be singular at and/or .

Monotone Positive Solutions for an Elastic Beam Equation with Nonlinear Boundary Conditions

This paper is concerned with the existence of monotone positive solutions for an elastic beam equation with nonlinear boundary conditions. By applying monotone iteration method, we not only obtain the existence of monotone positive solutions but also establish iterative schemes for approximating the solutions. It is worth mentioning that these iterative schemes start off with zero function or quadratic function, which is very useful and feasible for computational purpose. An example is also included to illustrate the main results obtained.

Existence and Iteration of Monotone Positive Solution of BVP for an Elastic Beam Equation

This paper is concerned with the existence of monotone positive solution of boundary value problem for an elastic beam equation. By applying iterative techniques, we not only obtain the existence of monotone positive solution but also establish iterative scheme for approximating the solution. It is worth mentioning that the iterative scheme starts off with zero function, which is very useful and feasible for computational purpose. An example is also included to illustrate the main results.

On the Dirichlet problem for a nonlinear elastic beam equation

Using a direct variational approach with no global growth conditions on the nonlinear term, we consider the existence of solutions and their dependence on a functional parameter for the fourth order Dirichlet problem connected with the elastic beam equation. We investigate also the existence of an optimal process for such an optimal control problem in which the dynamics is described by the beam equation.

A sum operator equation and applications to nonlinear elastic beam equations and Lane–Emden–Fowler equations

This paper is concerned with an operator equation A x + B x + C x = x on ordered Banach spaces, where A is an increasing α-concave operator, B is an increasing sub-homogeneous operator and C is a homogeneous operator. The existence and uniqueness of its positive solutions is obtained by using the properties of cones and a fixed point theorem for increasing general β-concave operators. As applications, we utilize the fixed point theorems obtained in this paper to study the existence and uniqueness of positive solutions for two classes nonlinear problems which include fourth-order two-point boundary value problems for elastic beam equations and elliptic value problems for Lane–Emden–Fowler equations.

A monotone iterative technique for solving the bending elastic beam equations

This paper discusses the solvability of the fourth-order boundary value problem u ( 4 ) = f ( t , u , u ″ ) , 0 ⩽ t ⩽ 1 , u ( 0 ) = u ( 1 ) = u ″ ( 0 ) = u ″ ( 1 ) = 0 , which models a statically bending elastic beam whose two ends are simply supported, where f : [ 0 , 1 ] × R 2 → R is continuous. We build a maximum principle for the corresponding linear equation. Using this maximum principle, we develop a monotone iterative technique in the presence of lower and upper solutions to solve the nonlinear equation and obtain some existence and uniqueness results.

Positive solutions to a class of elastic beam equations with semipositone nonlinearity

Positive solutions to a class of elastic beam equations with semipositone nonlinearity

Existence of Positive Solutions of a Discrete Elastic Beam Equation

Let T be an integer with T ≥ 5 and let . We consider the existence of positive solutions of the nonlinear boundary value problems of fourth‐order difference equations Δ4u(t − 2) − ra(t)f(u(t)) = 0, , u(1) = u(T + 1) = Δ2u(0) = Δ2u(T) = 0, where r is a constant, is continuous. Our approaches are based on the Krein‐Rutman theorem and the global bifurcation theorem.

Necessary and sufficient condition for the existence of positive solutions of a coupled system for elastic beam equations

In this work, by using fixed point theorem and monotone iterative technique, the author establishes a necessary and sufficient condition of the existence of positive solutions for a class of nonlinear singular elastic beam differential system under reasonable conditions. Moreover, some uniqueness results of positive solution and the estimate of convergent rate of the iterative sequence of solution are obtained.

Successively Iterative Technique of a Classical Elastic Beam Equation with Carathéodory Nonlinearity

In this paper, we consider an elastic beam equation where the nonlinear term is a Caratheodory function and the boundary condition is nonhomogeneous. We construct an iterative sequence by the help of monotonic technique and prove that the sequence approximates successively to the solution of the equation under suitable assumptions.

On the optimal control problem governed by the nonlinear elastic beam equation

We begin with using a dual variational method in proving the existence of solutions to a beam equation with Dirichlet boundary conditions and with a non-monotone nonlinear term. Later we investigate the dependence on a control parameter for the state equation. We find the optimal process for the optimal control problem in which the dynamics is governed by the nonlinear beam equation.

Existence and iteration of monotone positive solutions for an elastic beam equation with a corner

This paper is concerned with the existence of monotone positive solutions for an elastic beam equation with a corner. The boundary conditions mean that the beam is embedded at one end and fastened with a sliding clamp at the other end. By applying monotone iterative techniques, we not only obtain the existence of positive solutions, but also establish iterative schemes for approximating the solutions.

On the nonlinear elastic beam equation

We consider the existence of solutions and the continuous dependence on a parameter results for fourth-order Dirichlet problem connected with the elastic beam equation. We introduce a dual variational method.

A boundary value problem for fourth-order elastic beam equations

Multiplicity results for a fourth-order nonlinear boundary value problem are presented. The proof of the main result is based on the critical point theory.

Existence and multiplicity of positive solutions to a singular elastic beam equation rigidly fixed at both ends

The purpose of this paper is to study the existence and multiplicity of positive solutions for the fourth-order two-point boundary value problem u ( 4 ) ( t ) = λ f ( t , u ( t ) ) , 0 < t < 1 , u ( 0 ) = u ( 1 ) = u ′ ( t ) = u ′ ( 1 ) = 0 , where nonlinear term f ( t , u ) may be singular at t = 0 , t = 1 and u = 0 . In order to achieve the goal Guo–Krasnosel’skii fixed point theorem of cone expansion–compression type is applied. In mechanics, the problem describes the deflection of an elastic beam rigidly fixed at both ends.

Positive solutions of a nonlinear elastic beam equation rigidly fastened on the left and simply supported on the right

The purpose of this paper is to establish several local existence theorems concerned with n positive solutions for a fourth-order two-point boundary value problem, where n is an arbitrary positive integer and the nonlinear term may be singular. In mechanics, the problem describes deflection of an elastic beam rigidly fastened on the left and simply supported on the right.

The periodic boundary value problem for semilinear elastic beam equations: The resonance case

This paper discusses the existence of generalized solutions to periodic boundary value problems for semilinear elastic beam equations under a resonance condition. The argument presented makes use of the global inverse theorem and Galerkin’s method.